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A New Boundary Element for Plane Elastic Problems Involving Cracks and Holes

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Abstract

By applying the new boundary integral formulation proposed recently by Chau and Wang (1997) for two-dimensional elastic bodies containing cracks and holes, a new boundary element method for calculating the interaction between cracks and holes is presented in this paper. Singular interpolation functions of order r-1/2 (where r is the distance measured from the crack tip) are introduced for the discretization of the crack near the crack tips, such that stress singularity can be modeled appropriately. A nice feature for our implementation is that singular integrands involved at the element level are integrated analytically. For each of the hole boundaries, an additional unknown constant is introduced such that the displacement compatibility condition can be satisfied exactly by the complex boundary function H(t), which is a combination of the traction and displacement density. Another nice feature of the present formulation is that the stress intensity factors (both K_I and K_II) at crack tips are expressed in terms of the nodal unknown of H(t) exactly, and no extrapolation of numerical data is required. To demonstrate the accuracy of the present boundary element method, various crack problems are considered: (i) the Griffith crack problem, (ii) the interaction problem between a circular hole and a straight crack subject to both far field tension and compression, and (iii) the interaction problem between a circular hole and a kinked crack subject to far field uniaxial tension. Excellent agreement with existing results is observed for the first two problems and also for the last problem if the crack-hole interaction is negligible.

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References

  • Blandford, G.E., Ingraffea, A.R. and Liggett, J.A. (1981). Two-dimensional stress intensity factor computations using the boundary element method. International Journal for Numerical Methods in Engineering 17, 387–404.

    Article  MATH  Google Scholar 

  • Carpenter, W.C. (1984). A collocation procedure for determining fracture mechanics parameters at a corner. International Journal of Fracture 24, 255–266.

    Article  Google Scholar 

  • Chau, K.T. and Wang, Y.B. (1997). A new boundary integral formulation for two-dimensional elastic bodies containing cracks and holes. Submitted to International Journal of Solids and Structures.

  • Chou, S.I. (1997). Stress field around holes in antiplane shear using complex variable boundary element method. Journal of Applied Mechanics 64, 432–435.

    MATH  Google Scholar 

  • Crouch, S.L. (1976). Solution of plane elasticity problems by the displacement discontinuity method. International Journal for Numerical Methods in Engineering 10, 301–343.

    Article  MATH  MathSciNet  Google Scholar 

  • England, A.H. (1971a). On stress singularities in linear elasticity. International Journal of Engineering Science 9, 571–585.

    Article  MATH  Google Scholar 

  • England, A.H. (1971b). Complex Variable Methods in Elasticity. Wiley-Interscience, London, England.

    MATH  Google Scholar 

  • Griffith, A.A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London A211, 163–197.

    ADS  Google Scholar 

  • Hong, C.-C. and Stern, M. (1978). The computation of stress intensity factors in dissimilar materials. Journal of Elasticity 8, 21–34.

    Article  MATH  Google Scholar 

  • Hromadka, T.V. II (1984). The Complex Variable Boundary Element Method. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Hromadka, T.V. II and Lai, C. (1986). The Complex Variable Boundary Element Method in Engineering Analysis. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Isida, M. (1970). On the determination of stress intensity factors for some common structural problems. Engineering Fracture Mechanics 2, 61–79.

    Article  Google Scholar 

  • Kitagawa, H., Yuuki, R. and Ohira, T. (1975). Crack-morphological aspects in fracture mechanics. Engineering Fracture Mechanics 7, 515–529.

    Article  Google Scholar 

  • Lazzarin, P. and Tovo R. (1996). A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches. International Journal of Fracture 78, 3–19.

    Article  Google Scholar 

  • Linkov, A.M. and Mogilevskaya, S.G. (1990). Hypersingular integrals in plane problems of the theory of elasticity. Journal of Applied Mathematics and Mechanics (PMM) 54, 93–99.

    Article  MATH  MathSciNet  Google Scholar 

  • Linkov, A.M. and Mogilevskaya, S.G. (1994). Complex hypersingular integrals and integral equations in plane elasticity. Acta Mechanica 105, 189–205.

    Article  MATH  MathSciNet  Google Scholar 

  • Linkov, A.M., Mogilevskaya, S.G. and Napier, J.A.L. (1997). Multiple interacting curvilinear crack problems: A method of solution and numerical results. International Journal of Rock Mechanics and Mining Science 34, Paper No. 218.

    Google Scholar 

  • Lu, J.K. (1995). Complex Variable Methods in Plane Elasticity. World Scientific, Singapore.

    MATH  Google Scholar 

  • Mews, H. (1987). Calculation of stress intensity factors for various crack problems with the boundary element method. Proceedings of the 9th International Conference On Boundary Elements 2, 259–278.

    Google Scholar 

  • Milne-Thomson, L.M. (1968). Plane Elastic Systems. Springer-Verlag, Berlin, Germany.

    MATH  Google Scholar 

  • Murakami, Y. (1993). Stress Intensity Factors Handbook 3, Pergamon Press, Oxford, England.

    Google Scholar 

  • Muskhelishvili, N.I. (1975). Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff International, Leyden, Netherlands.

    MATH  Google Scholar 

  • Muskhelishvili, N.I. (1977). Singular Integral Equations. Noordhoff International, Leyden, Netherlands.

    Google Scholar 

  • Pan, E. and Amadei, B. (1996). Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method. International Journal of Fracture 77, 161–174.

    Article  Google Scholar 

  • Parker, A.P. (1981). The Mechanics of Fracture and Fatigue. E & F.N. Spon Ltd, London.

    MATH  Google Scholar 

  • Portela, A., Aliabadi, M.H. and Rooke, D.P. (1992). The dual boundary element method: effective implementation for crack problems. International Journal for Numerical Methods in Engineering 33, 1269–1287.

    Article  MATH  Google Scholar 

  • Sih, G.C. (1973). Handbook of Stress-Intensity Factors. Lehigh University, Bethlehem, Pennsylvania, USA.

    Google Scholar 

  • Sirtori, S., Maier, G., Novati, G. and Miccoli, S. (1992). A Galerkin symmetric boundary-element method in elasticity: Formulation and Implementation. International Journal for Numerical Methods in Engineering 35, 255–282.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Snyder, M.D. and Cruse, T.A. (1975). Boundary-integral equation analysis of cracked anisotropic plates. International Journal of Fracture 11, 315–328.

    Article  Google Scholar 

  • Sokolnikoff, I.S. (1956). Mathematical Theory of Elasticity. 2nd Ed., McGraw-Hill, New York.

    MATH  Google Scholar 

  • Timoshenko, S.P. and Goodier, J.N. (1951). Theory of Elasticity. 3rd Edition, McGraw Hill, New York.

    Google Scholar 

  • Wang, Y.B. (1993a). A new boundary integral equation method of three-dimensional crack analysis. International Journal of Fracture 63, 317–328.

    Article  ADS  Google Scholar 

  • Wang, Y.B. (1993b). A boundary integral equation method of the axisymmetric problem of an external circular crack. Journal of Lanzhou University 29, 19–24 (in Chinese).

    MATH  Google Scholar 

  • Wang, Y.B. and Chen, W.J. (1993). Interaction of two equal coplanar square cracks in three-dimensional elasticity. International Journal of Solids and Structures 30, 3315–3320.

    Article  MATH  Google Scholar 

  • Weaver, J. (1977). Three-dimensional crack analysis. International Journal of Solids and Structures 13, 321–330.

    Article  MATH  Google Scholar 

  • Wu, K.-C., Chiu, Y.-T. and Hwu, Z.-H. (1992). A new boundary integral equation formulation for linear elastic solids. Journal of Applied Mechanics 59, 344–348.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, Lanzhou University, Lanzhou, People's Republic of China

    Y.B. Wang

  2. Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    K.T. Chau

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  1. Y.B. Wang
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  2. K.T. Chau
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Correspondence to K.T. Chau.

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Wang, Y., Chau, K. A New Boundary Element for Plane Elastic Problems Involving Cracks and Holes. International Journal of Fracture 87, 1–20 (1997). https://doi.org/10.1023/A:1007469816603

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